Dependence and Independence of set of functionals

Question

Let $X$ be a Banach space on over $\mathbb{R}$ and let $\psi$ and $\phi$ be two linear functionals defined on $X$. Then, if $\ker(\phi)\subset \ker(\psi)$, can we have that the set $\{\phi, \psi\}$ is linearly dependent?

Answer 1

They are always linearly dependent. That's trivial if one of them is $0$. Otherwise, take $v\in X$ such that $\phi(v)\neq0$. Let $\lambda=\psi(v)$ and consider the map $\psi-\lambda\phi$. Then, if $x\in X$, $x=x-\phi(x)v+\phi(x)v$, and $x-\phi(x)v\in\ker\phi$. Therefore, $x-\phi(x)v\in\ker\psi$ and\begin{align}\psi(x)&=\psi\bigl(x-\phi(x)v\bigr)+\psi\bigl(\phi(x)v\bigr)\\&=\lambda\phi(x).\end{align}So, $\psi-\lambda\phi=0$ and therefore $\psi$ and $\phi$ are linearly dependent.